Physics C: Electricity and Magnetism 2007 Free-Response Questions

AP

Physics C: Electricity and Magnetism 2007 Free-Response Questions

The College Board: Connecting Students to College Success

The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the association is composed of more than 5,000 schools, colleges, universities, and other

educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,500 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT^®, the PSAT/NMSQT^®, and the Advanced Placement Program^® (AP^®). The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities, and concerns.

© 2007 The College Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, SAT, and the

TABLE OF INFORMATION FOR 2006 and 2007 CONSTANTS AND CONVERSION FACTORS

1 unified atomic mass unit, ²⁷

2

1 u 1.66 10 kg 931 MeV c

= ¥ -

=

Proton mass, m_p=1.67¥10^-27kg Neutron mass, m_n=1.67¥10^-27kg Electron mass, m_e= 9.11¥10^-31kg Electron charge magnitude, e=1.60¥10^-19C

Avogadro’s number, N₀ = 6.02¥10 mol^{23 -1} Universal gas constant, R =8.31 J (mol K)ⁱ

Boltzmann’s constant, k_B =1.38¥10^-23J /K Speed of light, c =3.00¥10 m s⁸ Planck’s constant, ³⁴

15

6.63 10 J s 4.14 10 eV s

h ^-

-

= ¥

= ¥

i i

25 3

1.99 10 J m 1.24 10 eV nm hc = ¥ ^-

= ¥

i i

Vacuum permittivity, ⑀₀=8.85¥10^-12C² N m^{i 2} Coulomb’s law constant, k =1 4p⑀₀ = 9.0¥10 N m⁹ i ² C²

Vacuum permeability, m₀ = 4p¥10^-7(T m) Aⁱ Magnetic constant, k¢ = ^m₀/4^p =10^-7(T m) A

◊

Universal gravitational constant, G =6.67¥10^-11m³ kg si ² Acceleration due to gravity

at Earth’s surface, g =9.8 m s²

1 atmosphere pressure, ^{5 2}

5

1 atm 1.0 10 N m 1.0 10 Pa

= ¥

= ¥

1 electron volt, 1 eV =1.60¥10^-19J

UNITS Name Symbol meter m kilogram kg second s ampere A kelvin K mole mol hertz Hz newton N pascal Pa joule J watt W coulomb C volt V

ohm W

henry H farad F tesla T degree

Celsius ∞ C electron- volt eV

PREFIXES Factor Prefix Symbol

109 giga G 106 mega M 103 kilo k 10^-2 centi c 10^-3 milli m 10^-6 micro m 10^-9 nano n 10^-12 pico p

VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON

ANGLES

q sin q cos q tan q

0 0 1 0

30 1/2 3/2 3/3 37 3/5 4/5 3/4

45 2/2 2/2 1 53 4/5 3/5 4/3

60 3/2 1/2 3

90 1 0

•

The following conventions are used in this examination.

I. Unless otherwise stated, the frame of reference of any problem is assumed to be inertial.

II. The direction of any electric current is the direction of flow of positive charge (conventional current).

III. For any isolated electric charge, the electric potential is defined as zero at an infinite distance from the charge.

ADVANCED PLACEMENT PHYSICS C EQUATIONS FOR 2006 and 2007

MECHANICS ELECTRICITY AND MAGNETISM

0 at

u = u +

2

0 0

1 x = x +u t + 2at

( )

2 2

0 2a x x0

u = u + -

net m

 =F F = a d

= dtp F

dt D

=

Ú

J F p

= m

p v

Ffric £ mN W =

Ú

1 2

K = 2mu P dW

= dt P = F vⴢ

Ug mgh

D =

2 2

ac r

r

u w

= =

= ¥r F t

net I

 =t t = a

2 2

I =

Ú

cm = Âm Âm

r r

a = acceleration F = force f = frequency h = height

I = rotational inertia J = impulse

K = kinetic energy k = spring constant

= length

L = angular momentum m = mass

N = normal force P = power p = momentum r = radius or distance r = position vector T = period

t = time

U = potential energy u = velocity or speed W = work done on a system x = position

m = coefficient of friction q = angle

t = torque w = angular speed

a = angular acceleration

s = -k

F x

1 2

s 2

U = kx

2 1

T f

p

= w =

s 2 m

T = p k

2 1 2 0

1 4 F q q

p r

= ⑀

= Fq E

0

d Q

∑ =

Ú

E dV

= - dr

0

1 4

i i i

V q p r

= _⑀

Â

1 2 0

1

E 4 U qV q q

p r

= = ⑀

C Q

= V

0A

C d

= k⑀

p i

i

C =

Â

1 1

s i i

C =

Â

= dt

1 1 2

2 2

Uc = QV = CV

R A

= r

= r

E J

I = Ne Aud

A = area

B = magnetic field C = capacitance d = distance E = electric field

e

J = current density L = inductance

= length

n = number of loops of wire per unit length

N = number of charge carriers per unit volume P = power

Q = charge q = point charge R = resistance r = distance t = time

U = potential or stored energy V = electric potential

u = velocity or speed r = resistivity f_m = magnetic flux k = dielectric constant

d m0I

∑ =

Ú

d I d

r

0

4 3

m p

= ¥r

B ᐉ

=

Ú

F ᐉ B

s 0

B = m nI

ADVANCED PLACEMENT PHYSICS C EQUATIONS FOR 2006 and 2007

GEOMETRY AND TRIGONOMETRY CALCULUS

Rectangle A = bh Triangle 1

A = 2bh Circle A = pr² C = 2pr Parallelepiped V = wh Cylinder V = pr²

S = 2pr +2pr² Sphere

4 ³ V = 3pr S = 4pr² Right Triangle a² +b² = c² sin a

q = c cos b

q = c tan a

q = b

A = area

C = circumference V = volume S = surface area b = base h = height

= length w = width r = radius

c a

b q 90°

d f d f du dx = du dx

( )

d x nx

dx = ^-

( )

d e e

dx =

(

)

d x

dx = x

(

)

d x x

dx =

(

)

d x x

dx = -

1 1

, 1

1

n n

x dx x n

n ⁺

= + π -

Ú

x x

e dx = e

Ú

dx ln x = x

Ú

cosx dx =sinx

Ú

sinx dx = -cosx

Ú

2007 AP

PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS

PHYSICS C:

Time—45 minutes 3 Questions

Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions, which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in the pink booklet in the spaces provided after each part, NOT in this green insert.

E&M 1.

A student sets up the circuit above in the lab. The values of the resistance and capacitance are as shown, but the constant voltage

e

(a) Using the data above, calculate the battery voltage

e

(b) Calculate the voltage across the capacitor at time t = 4.0 s.

(c) Calculate the charge on the capacitor at t = 4.0 s.

2007 AP

PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS

© 2007 The College Board. All rights reserved.

Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).

GO ON TO THE NEXT PAGE.

(d) On the axes below, sketch a graph of the charge on the capacitor as a function of time.

(e) Calculate the power being dissipated as heat in the resistor at t = 4.0 s.

(f) The capacitor is now discharged, its dielectric of constant k = is replaced by a dielectric of constant 1 k = , 3 and the procedure is repeated. Is the amount of charge on one plate of the capacitor at t = 4.0 s now greater than, less than, or the same as before? Justify your answer.

_____Greater than _____Less than _____The same

2007 AP

PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS

E&M 2.

In the figure above, a nonconducting solid sphere of radius a with charge +Q uniformly distributed throughout its volume is concentric with a nonconducting spherical shell of inner radius 2a and outer radius 3a that has a charge -Q uniformly distributed throughout its volume. Express all answers in terms of the given quantities and fundamental constants.

(a) Using Gauss’s law, derive expressions for the magnitude of the electric field as a function of radius r in the following regions.

i. Within the solid sphere ( r < ) a

ii. Between the solid sphere and the spherical shell (a < <r 2a) iii. Within the spherical shell ( 2a < <r 3a)

iv. Outside the spherical shell (r >3a)

(b) What is the electric potential at the outer surface of the spherical shell (r = 3a)? Explain your reasoning.

(c) Derive an expression for the electric potential difference V_X -V_Y between points X and Y shown in the figure.

2007 AP

PHYSICS C: ELECTRICITY AND MAGNETISM FREE-RESPONSE QUESTIONS

© 2007 The College Board. All rights reserved.

Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).

E&M 3.

In the diagram above, a nichrome wire of resistance per unit length l is bent at points P and Q to form horizontal conducting rails that are a distance L apart. The wire is placed within a uniform magnetic field of magnitude B pointing into the page. A conducting rod of negligible resistance, which was aligned with end PQ at time t = 0, slides to the right with constant speed u and negligible friction. Express all algebraic answers in terms of the given

quantities and fundamental constants.

(a) Indicate the direction of the current induced in the circuit.

______Clockwise ______Counterclockwise Justify your answer.

(b) Derive an expression for the magnitude of the induced current as a function of time t.

(c) Derive an expression for the magnitude of the magnetic force on the rod as a function of time.

(d) On the axes below, sketch a graph of the external force F_ext as a function of time that must be applied to the rod to keep it moving at constant speed while in the field. Label the values of any intercepts.

(e) The force pulling the rod is now removed. Indicate whether the speed of the rod increases, decreases, or remains the same.

______Increases _____Decreases ______Remains the same Justify your answer.

END OF EXAM